Average Error: 30.9 → 0.2
Time: 21.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} - x\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} - x\right)\right)\right)
double f(double x) {
        double r14519812 = x;
        double r14519813 = r14519812 * r14519812;
        double r14519814 = 1.0;
        double r14519815 = r14519813 - r14519814;
        double r14519816 = sqrt(r14519815);
        double r14519817 = r14519812 + r14519816;
        double r14519818 = log(r14519817);
        return r14519818;
}
↓
double f(double x) {
        double r14519819 = x;
        double r14519820 = -0.125;
        double r14519821 = r14519820 / r14519819;
        double r14519822 = r14519819 * r14519819;
        double r14519823 = r14519821 / r14519822;
        double r14519824 = 0.5;
        double r14519825 = r14519824 / r14519819;
        double r14519826 = r14519825 - r14519819;
        double r14519827 = r14519823 - r14519826;
        double r14519828 = r14519819 + r14519827;
        double r14519829 = log(r14519828);
        return r14519829;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]

  2. Simplified30.9

    \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}\]

  3. Taylor expanded around inf 0.2

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]

  4. Simplified0.2

    \[\leadsto \log \left(x + \color{blue}{\left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} - x\right)\right)}\right)\]

  5. Final simplification0.2

    \[\leadsto \log \left(x + \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} - x\right)\right)\right)\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))